\section{Category of Sheaves}

\section{Points of sites}
In the theory of sheaves on topological spaces, one sees importance of the stalks of sheaves in determining the behaviour of sheaves on open subset of the topological space \cite[Prop 1.1 P 63, Def 1.2 P 64]{har77}. In this we will recall points of topoi (and focus on points of sites) and stalks of (pre-)sheaves on points of sites. Here, we will be mainly following \cite{SGA4IV} and \cite{Mac 92}.\\

We start with recalling that for a topological space $X$, and any point $x\in X$, one have the continuous map $i:\{x\}\rightarrow X$ which induce an adjunction $i^{\ast} \stackrel{\varphi}{\vdash} i_{\ast}$ (as sheaves of sets), where $i^{\ast}$ is left-exact, hence a morphism of topoi $( i_{\ast},i^{\ast},\varphi):\Shv(\{x\})\rightarrow \Shv(X)$ - as seen later . The topological space $\{x\}$ has only two open subsets $\emptyset$ and $\{x\}$, and for any sheaf of sets $\bcF$ on $\{x\}$, $\bcF(\emptyset)$ is a singleton, then $\bcF$ is determined up to isomorphism by the choice of the set $\bcF(\{x\})$. Hence, one can see easily that $\Shv(\{x\}) \cong \Sets$. This case will be the motivation for the definition of points on topoi.

In order to study point in the general setting, we recall basic definitions of topoi.

\begin{definition}
A topos is category equivalent to the category of sheaves of sets on some small site.
\end{definition}
We saw that continuous morphisms of sites $f:\bcC_{\tau}\rightarrow \bcD_{\sigma}$ induce an adjunction $f^{\ast}\vdash f_{\ast}$ with $f^{\ast}$ is left-exact. That, in turn motivate the following definition:
\begin{definition}

\end{definition}



\begin{definition}[Points of sites]
Let $\bcC_{\tau}$ be a site, 

\end{definition}








Here, we avoided dealing topoi, butwe should mention that points are defined in a more general sense, they are defined similarly on toposes (categories equivalent to the category of sheaves of sets on some small site).


